1. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-1 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Chapter 12 Network Models Prepared by Lee Revere and John Large

2. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-2 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Learning Objectives Students will be able to: 1.Connect all points of a network while minimizing total distance using the minimal-spanning tree technique. 2.Determine the maximum flow through a network using the maximal-flow technique. 3.Find the shortest path through a network using the shortest-route technique. 4.Understand the important role of software in solving network problems.

3. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-3 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Chapter Outline 12.1 12.1Introduction 12.2 12.2Minimal-Spanning Tree Technique 12.3 12.3Maximal-Flow Technique 12.4 12.4Shortest-Route Technique

4. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-4 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Introduction  The presentation will cover three network models that can be used to solve a variety of problems: 1.the minimal-spanning tree technique, 2.the maximal-flow technique, and 3.the shortest-route technique.

5. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-5 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Minimal-Spanning Tree Technique Definition: The minimal-spanning tree technique determines the path through the network that connects all the points while minimizing total distance. For example: If the points represent houses in a subdivision, the minimal spanning tree technique can be used to determine the best way to connect all of the houses to electrical power, water systems, etc. in a way that minimizes the total distance or length of power lines or water pipes.

6. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-6 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 The Maximum Flow Technique Definition: The maximal-flow technique finds the maximum flow of any quantity or substance through a network. For example: This technique can determine the maximum number of vehicles (cars, trucks, etc.) that can go through a network of roads from one location to another.

7. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-7 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Shortest Route Technique Definition: Shortest route technique can find the shortest path through a network. For example: This technique can find the shortest route from one city to another through a network of roads.

8. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-8 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Minimal-Spanning Tree Steps 1.Selecting any node in the network. 2.Connecting this node to the nearest node minimizing the total distance. 3.Finding and connecting the nearest unconnected node.  If there is a tie for the nearest node, one can be selected arbitrarily.  A tie suggests that there may be more than one optimal solution. 4.Repeating the third step until all nodes are connected.

9. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-9 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Minimal-Spanning Tree Technique Solving the network for Melvin Lauderdale construction  Start by arbitrarily selecting node 1.  Since the nearest node is the third node at a distance of 2 (200 feet), connect node 1 to node 3.  Shown in Figure 12.2 (2 slides hence)  Considering nodes 1 and 3, look for the next-nearest node.  This is node 4, which is the closest to node 3 with a distance of 2 (200 feet).  Once again, connect these nodes (Figure 12.3a (3 slides hence).

10. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-10 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Figure 12.1: Network for Lauderdale Construction

11. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-11 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Figure 12.2: First Iteration Lauderdale Construction

12. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-12 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Fig 12.3a: Second Iteration

13. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-13 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Fig 12.3b: Third Iteration

14. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-14 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Summarize: Minimal- Spanning Tree Technique Step 1:Select node 1 Step 2:Connect node 1 to node 3 Step 3:Connect the next nearest node Step 4:Repeat the process  The total number of iterations to solve this example is 7.  This final solution is shown in the following slide.

15. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-15 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Fig 12.5b: Third Iteration

16. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-16 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Final Solution to the Minimal-Spanning Tree Example  Nodes 1, 2, 4, and 6 are all connected to node 3. Node 2 is connected to node 5.  Node 6 is connected to node 8, and node 8 is connected to node 7.  All of the nodes are now connected.  The total distance is found by adding the distances for the arcs used in the spanning tree.  In this example, the distance is: 2 + 2 + 3 + 3 + 3 + 1 + 2 = 16 (or 1,600 feet).

17. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-17 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Maximal-Flow Technique maximal-flow technique  The maximal-flow technique allows the maximum amount of a material that can flow through a network to be determined.  For example:  It has been used to find the maximum number of automobiles that can flow through a state highway system.  An example:  Waukesha is in the process of developing a road system for downtown.  City planners would like to determine the maximum number of cars that can flow through the town from west to east.  The road network is shown in Figure 12.6 (next slide).

18. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-18 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Road Network for Waukesha  Traffic can flow in both directions.

19. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-19 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Maximal-Flow Technique (continued) The Four Maximal-Flow Technique Steps: 1.Pick any path from the start (source) to the finish (sink) with some flow.  If no path with flow exists, then the optimal solution has been found. 2.Find the arc on this path with the smallest flow capacity available.  Call this capacity C.  This represents the maximum additional capacity that can be allocated to this route. 3.For each node on this path, decrease the flow capacity in the direction of flow by the amount C.  For each node on this path, increase the flow capacity in the reverse direction by the amount C. 4.Repeat these steps until an increase in flow is no longer possible.

20. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-20 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Solving the Waukesha Example  Start by arbitrarily picking the path 1–2–6, at the top of the network.  What is the maximum flow from west to east? It is 2 because only 2 units (200 cars) can flow from node 2 to node 6.  Now we adjust the flow capacities (Figure 12.7). As you can see, we subtracted the maximum flow of 2 along the path 1–2–6 in the direction of the flow (west to east) and added 2 to the path in the direction against the flow (east to west).  The result is the new path in Figure 12.7 (next slide).

21. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-21 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Capacity Adjustment 1 2 6 1 2 2 3 East Point West Point Add 2 Subtract 2 Iteration 1 1 2 6 3 0 4 1 East Point West Point New path

22. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-22 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Solving the Waukesha Example  The New Path reflects the new relative capacity at this stage.  The flow number by any node represents two factors.  One factor is the flow that can come from that node.  The second factor is flow that can be reduced coming into the node.  The number 1 by node 1 indicates that 100 cars can flow from node 1 to node 2. 1 2 6 3 0 4 1 East Point West Point New path

23. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-23 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Solving the Waukesha Example  The number 0 by node 2 on the path from node 2 to node 6 indicates that 0 cars can flow from node 2 to node 6. 1 2 6 3 0 4 1 East Point West Point New path  The number 4 by node 6, on the path from node 6 to node 2, indicates that we can reduce the flow into node 6 by 2 (or 200 cars) and that there is a capacity of 2 (or 200 cars) that can come from node 6.  These two factors total 4.

24. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-24 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Solving the Waukesha Example  On the path from node 2 to node 1, the number 3 by node 2 shows that we can reduce the flow into node 2 by 2 (or 200 cars) and that there is a capacity of 1 (or 100 cars) from node 2 to node 1 1 2 6 3 0 4 1 East Point West Point New path  At this stage, there is a flow of 200 cars through the network from node 1 to node 2 to node 6.  The new relative capacity reflects this.

25. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-25 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Repeat the Process  Now, repeat the process by picking another path with existing capacity.  Can arbitrarily pick path 1–2–4–6.  The maximum capacity along this path is 1.  In fact, the capacity at every node along this path (1–2–4–6) going from west to east is 1.  Remember, the capacity of branch 1–2 is now 1 because 2 units (200 cars per hour) are now flowing through the network.  So, need to increase the flow along path 1–2–4–6 by 1 and adjust the capacity flow (see next slide).

26. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-26 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Fig-12.8a: Second Iteration for Waukesha 1 2 4 6 1 3 1 11 1 Subtract 1 Old Path Add 1

27. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-27 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Fig-12.8b: Second Iteration for Waukesha 1 2 4 3 5 6 4 0 0 20 2 4 0 6 1 2 3 0 10 2 0 0 1 East Point West Point New Network

28. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-28 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458  Now, there is a flow of 3 units (300 cars):  200 cars per hour along path 1–2–6 plus  100 cars per hour along path 1–2–4–6  Can the flow be further increased?  Yes, along path 1–3–5–6.  This is the bottom path.  The maximum flow is 2 because this is the maximum from node 3 to node 5.  The increased flow along this path is shown in the next slide. Continuing the Process

29. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-29 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Third Iteration 1 2 4 3 5 6 4 0 0 20 2 4 0 6 1 2 3 0 10 2 0 0 1 East Point West Point Add 2 Subtract 2

30. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-30 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Again, repeat the process.  Try to find a path with any unused capacity through the network.  Carefully checking the third iteration in the last slide reveals that there are no more paths from node 1 to node 6 with unused capacity,  even though several other branches in the network do have unused capacity.  The final network appears on the next slide. Continuing the Process

31. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-31 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Final Iteration 1 2 4 3 5 6 4 0 0 20 2 4 2 4 3 0 3 0 10 2 0 0 1 East Point West Point New Path Path = 1, 3, 5, 6

32. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-32 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Final Network Flow Cars per Hour (Cars per Hour) PATHFLOW PATH FLOW 1-2-6200 1-2-4-6100 1-3-5-6200 Total Total =500 The maximum flow of 500 cars per hour is summarized in the following table:

33. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-33 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 The Shortest-Route Technique shortest-route technique  The shortest-route technique minimizes the distance through a network.  The shortest-route technique finds how a person or item can travel from one location to another while minimizing the total distance traveled.  The shortest-route technique finds the shortest route to a series of destinations.

34. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-34 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example: From Ray’s Plant to Warehouse For example,  Every day, Ray Design, Inc., must transport beds, chairs, and other furniture items from the factory to the warehouse.  This involves going through several cities.  Ray would like to find the route with the shortest distance.  The road network is shown on the next slide.

35. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-35 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Roads from Ray’s Plant to Warehouse: Shortest-Route Technique (continued) 1 2 3 4 5 6 100 150 200 50 40 200 Warehouse Plant

36. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-36 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Steps of the Shortest- Route Technique 1.Find the nearest node to the origin (plant). Put the distance in a box by the node. 2.Find the next-nearest node to the origin (plant), and put the distance in a box by the node. In some cases, several paths will have to be checked to find the nearest node. 3.Repeat this process until you have gone through the entire network. The last distance at the ending node will be the distance of the shortest route.

37. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-37 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Ray Design: 1 st Iteration Shortest-Route Technique (continued)  The nearest node to the plant is node 2, with a distance of 100 miles.  Thus, connect these two nodes. 1 2 3 4 5 6 100 150 200 50 40 200 Warehouse Plant 100

38. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-38 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Ray Design: 2nd Iteration Shortest Route Technique (continued)  The nearest node to the plant is node 3, with a distance of 50 miles.  Thus, connect these two nodes. 1 2 3 4 5 6 100 150 200 50 40 200 100 150

39. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-39 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Ray Design: 3rd Iteration Shortest-Route Technique (continued)  The nearest node to the plant is node 5, with a distance of 40 miles.  Thus, connect these two nodes. 1 2 3 4 5 6 100 150 200 50 40 200 100 150 190

40. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 12-40 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 4 th and Final Iteration Shortest Route Technique (continued) 1 2 3 4 5 6 100 150 200 50 40 200 100 150 190 290 Total Shortest Route = 100 + 50 + 40 + 100 = 290 miles.